US7158670B1ExpiredUtility
Visually error-free color image representation
Est. expiryMay 21, 2019(expired)· nominal 20-yr term from priority
H04N 1/6058
48
PatentIndex Score
14
Cited by
19
References
20
Claims
Abstract
The present invention provides a method, apparatus and computer-readable medium for providing high quality substantially visually error-free conversion of color images. The method maps the color values to an expanded colorspace to represent substantially all colors without perceptible visual error and labels an image determined by the mapped color values as an expanded colorspace image. Thus, the present invention may be utilized to provide a linear color management system. In a 16 bit color data value system, transparency/opaqueness information may be stored, and premultiplied or premultiplied and normalized values may be used for blending.
Claims
exact text as granted — not AI-modified1. A method for providing a color space representation of color images in a color management system, comprising the steps of:
mapping RGB color data values representing an image in a first device into gamut expanded sRGB color data values of a gamut expanded sRGB color space; and
converting the gamut expanded sRGB color data values of the gamut expanded sRGB color space into RGB color data values representing an image in a second device, the RGB color data values of the first device being different from the RGB color data values of the second device and the physical appearance of the image in the first device being the same as the physical appearance of the image in the second device,
wherein the step of mapping includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value, wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to the gamut expanded sRGB color data values of the gamut expanded sRGB color space is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein the step of converting includes converting from the gamut expanded sRGB color data values of the gamut expanded sRGB color space to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.2126
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values with respect to a standard illuminant defined by CIE,
wherein the RGB color data values representing the image in the second device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
wherein m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.
2. The method of claim 1 wherein if the color data values in the gamut expanded sRGB color space lie outside a range of the RGB color data values of the second device, further including clipping the color data values for the second device.
3. The method of claim 2 , wherein mapping includes, where color data values of the first device have been represented using signed 16 bit values and 13 bits of decimal precision, clipping the 16 bit values below 0 and above 8192 to convert the 16 bit values to 8 bit values.
4. The method of claim 1 wherein the gamut expanded sRGB color space is linear in visual intensity.
5. The method of claim 1 wherein the gamut expanded sRGB color space comprises an XsRGB color space that includes at least the visible range of color values, and where selected, the gamut expanded sRGB color space includes an alpha channel for at least one of: transparency information and opaqueness information.
6. The method of claim 1 wherein the gamut expanded sRGB color space includes a color space defined by a gamut that extends into negative component values and beyond 1.0 when normalized to 1.0 in RGB.
7. The method of claim 1 wherein each color data value of the image in the first device uses a signed 16 bit integer and 13 bits are used as a decimal portion.
8. The method of claim 7 wherein 16 bit components R 16 , G 16 and B 16 are given by:
[
R
16
G
16
B
16
]
=
8192
×
[
R
0
G
0
B
0
]
where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value.
9. The method according to claim 1 , wherein the color data values of the first device are one of: non-premultiplied color data values, premultiplied color data values, and normalized numerically linear premultiplied color data values.
10. In a digitized image processing system in which an image digitizer outputs digital signals representing an image, a method for providing representation of color images from measured RGB color data values in a color management system, comprising the steps of:
mapping the measured RGB color data values to a gamut expanded sRGB color space, wherein the gamut expanded sRGB color space includes color data values beyond a reproduction range of a specific device and includes all colors in a humanly visible gamut; and
converting the gamut expanded sRGB color data values of the gamut expanded sRGB color space into RGB color data values representing an image in a destination device, the measured RGB color data values being different from the RGB color data values of the destination device and the physical appearance of the image output by the image digitizer being the same as the physical appearance of the image in the destination device,
wherein the step of mapping includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value,
wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to the gamut expanded sRGB color data values of the gamut expanded sRGB color space is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein the step of converting includes converting from the gamut expanded sRGB color data values of the gamut expanded sRGB color space to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.21262
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values with respect to a standard illuminant defined by CIE,
wherein the RGB color data values representing the image in the destination device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
where m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.
11. The method of claim 10 wherein the gamut expanded sRGB color space includes an XsRGB color space defined by a gamut that extends into negative component values and beyond 1.0 when normalized to 1.0 in RGB, and where selected, wherein the expanded sRGB color space includes an alpha channel for at least one of: transparency information and opaqueness information.
12. The method of claim 10 , wherein each measured RGB color data value uses a 16 bit integer and 13 bits are used as a decimal portion.
13. The method of claim 10 , wherein 16 bit components R 16 , G 16 , and B 16 of measured color data values are given by:
[
R
16
G
16
B
16
]
=
8192
×
[
R
0
G
0
B
0
]
where the R 0 , G 0 , B 0 values denote normalized numerically linear red, green and blue components for a color value.
14. The method of claim 10 , wherein if measured RGB color data values have been represented using signed 16 bit values with 13 bits of decimal precision, further including clipping the 16 bit values below 0 and above 8192 to convert the 16 bit values to 8 bit values.
15. The method of claim 10 , wherein the measured RGB color data values are one of: non-premultiplied color data values, premultiplied color data values, and normalized numerically linear premultiplied color data values.
16. A computer-readable medium having computer-executable instructions for performing the steps of:
mapping measured color data values to a gamut expanded sRGB color space, wherein the gamut expanded sRGB color space includes color data values beyond a reproduction range of a specific device and includes all colors in a humanly visible gamut; and
converting the gamut expanded sRGB color data values of the gamut expanded sRGB color space into RGB color data values representing an image in a destination device, the measured color data values being different from the RGB color data values of the destination device and the physical appearance of an image output by a digitizer device being the same as the physical appearance of the image in the destination device,
wherein the step of mapping includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value,
wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to the gamut expanded sRGB color data values of the gamut expanded sRGB color space is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein the step of converting includes converting from the gamut expanded sRGB color data values of the gamut expanded sRGB color space to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.21262
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values with respect to a standard illuminant defined by CIE,
wherein the RGB color data values representing the image in the destination device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
wherein m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.
17. In a digitized image processing system in which an image digitizer utilizes color image information to output RGB digital color signals representing a color image to an apparatus that uses the digital color signals to provide representation of a color image in a color management system, the apparatus comprising:
an expanded sRGB color space mapper, for mapping the digital color signals representing RGB color data values of the image digitizer to gamut expanded sRGB color space values; and
a processor for converting said gamut expanded sRGB color space values to RGB color space values representing an image in a destination peripheral device, the RGB color data values of the image digitizer being different from the RGB color data values of the destination peripheral device and the physical appearance of the image in the image digitizer being the same as the physical appearance of the image in the destination peripheral device,
wherein the expanded sRGB color space mapper for mapping includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value,
wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to the gamut expanded sRGB color data values is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein the step of converting includes converting from the gamut expanded sRGB color data values to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.21262
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values with respect to a standard illuminant defined by CIE,
wherein the RGB color data values representing the image in the destination peripheral device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
wherein m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.
18. A method for representing color images in a color management system in a gamut expanded sRGB color space and further representing at least one of super transparent and super opaque colors using an alpha channel, comprising the steps of:
representing RGB color data values of a source peripheral device as one of perceptually visible super transparent data values and perceptually visible super opaque data values in said gamut expanded sRGB color space; and
converting one of said perceptually visible super transparent data values and perceptually visible super opaque data values to RGB color data values of a destination peripheral device, the RGB color data values of the source peripheral device being different from the RGB color data values of the destination peripheral device and the physical appearance of an image represented by the RGB color data values in the source peripheral device being the same as the physical appearance of an image represented by the RGB color date values in the destination peripheral device,
wherein the step of representing includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value,
wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to gamut expanded sRGB color data values of the gamut expanded sRGB color space is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein the step of converting includes converting from the gamut expanded sRGB color data values of the gamut expanded sRGB color space to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.2126
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values respect to a standard illuminant defined by CIE,
wherein the RGB color data values representing the image in the destination peripheral device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
-
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
wherein m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.
19. A method for converting color data values comprising the steps of:
mapping color data values representing an image in a first device into color data values of a XsRGB color space; and
converting the XsRGB color data values into color data values representing an image in a second device,
wherein the step of mapping includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value,
wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to the XsRGB color space is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein the step of converting includes converting from the XsRGB color data values to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.2126
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values with respect to a standard illuminant defined by CIE,
wherein the color data values representing the image in the second device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
wherein m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.
20. A computer-readable medium comprising computer readable instructions that, when executed, cause a computer to perform a method for converting color data values, comprising:
mapping color data values representing an image in a first device into color data values of a XsRGB color space; and
converting the XsRGB color data values into color data values representing an image in a second device,
wherein the step of mapping includes utilizing multiplication of R 0 , G 0 , B 0 values by a predetermined matrix, where the R 0 , G 0 , and B 0 values denote normalized numerically linear red, green and blue components for a color value,
wherein XYZ values are obtained in accordance with the following:
(
X
Y
Z
1
)
=
M
-
1
(
R
0
G
0
B
0
1
)
,
where
M
-
1
=
[
n
XR
n
XG
n
XB
u
X
n
YR
n
YG
n
YB
u
Y
n
ZR
n
ZG
n
ZB
u
Z
0
0
0
1
]
,
wherein X, Y, and Z denote 1931 Commission Internationale de l'Eclairage XYZ values where Y has been normalized to 1,
wherein the XYZ values with respect to the D 65 standard are obtained in accordance with the following:
{
X
D65
=
x
D65
/
y
D65
=
0.9502
Y
D65
=
1.0
Z
D65
=
(
1.0
-
x
D65
-
y
D65
)
/
y
D65
=
1.0887
,
wherein a transformation matrix from XYZ to the XsRGB color space is obtained in accordance with the following:
M
w
=
M
D65
S
w
,
where
M
D65
=
[
3.2410
-
1.5374
-
0.4986
0
-
0.9692
1.8760
0.0416
0
0.0556
-
0.2040
1.0570
0
0
0
0
1
]
and
,
where
S
w
=
[
X
D65
/
X
w
0
0
0
0
1
0
0
0
0
Z
D65
/
Z
w
0
0
0
0
1
]
,
wherein X w and Z w denote corresponding X and Z values with respect to a standard illuminant defined by CIE,
wherein the step of converting includes converting from the XsRGB color data values to XYZ values based upon a transverse matrix obtained in accordance with the following:
M
w
-
1
=
S
w
-
1
M
D65
-
1
,
where
M
D65
-
1
=
[
0.4124
0.3576
0.1805
0
0.2126
0.7152
0.0722
0
0.0193
0.1192
0.9505
0
0
0
0
1
]
and
,
where
S
w
-
1
=
[
X
w
/
X
D65
0
0
0
0
1
0
0
0
0
Z
w
/
Z
D65
0
0
0
0
1
]
,
wherein the color data values representing the image in the second device are obtained in accordance with the following:
(
R
0
G
0
B
0
1
)
=
M
(
X
Y
Z
1
)
,
where
M
=
[
m
RX
m
RY
m
RZ
t
R
m
GX
m
GY
m
GZ
t
G
m
BX
m
BY
m
BZ
t
B
0
0
0
1
]
,
wherein m denotes a rotational part, t denotes a transitional part, n denotes an inverse rotational part, and u denotes an inverse transitional part.Cited by (0)
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